Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one quote per post.
Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of? Standard community wiki rules apply: one quote per post. 

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Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by immersing it in some soft liquid and let time pass: "I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance." Grothendieck, of course, always pioneered this approach, and considered for example that JeanPierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a very elegant way. 


"The question you raise, "how can such a formulation lead to computations?" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand  and it always turned out that understanding was all that mattered."  Grothendieck 


"There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms."  Eichler 


(Caveat for all of mine: I've not hunted down primary sources to check that they're properly attributed) "Manifolds are a bit like pornography: hard to define, but you know one when you see one." S. Weinberger 


"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics."  Weyl 


We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?"  GianCarlo Rota 


"The shortest path between two truths in the real domain passes through the complex domain."  Jacques Hadamard 


"The art of doing mathematics is finding that special case that contains all the germs of generality."  David Hilbert 


— Richard Hamming (1962)
– Richard W. Hamming, Introduction to applied numerical analysis, McGrawHill 1971, p.31. 


My favorite math quote will probably always be Paul Gordan's response to Hilbert's proof of his Basis Theorem: "This is not Mathematics. This is Theology." Along with his redaction after he came to accept the method: "I have convinced myself that even theology has its merits." 


Free translation: keep going, faith will come later. JeanleRond D'Alembert, to his students (quoted by Florian Cajori in A history of mathematics) 


A mathematician is a device for turning coffee into theorems. —Alfréd Rényi, but often attributed also to Paul Erdős 


Laurent Schwartz, Un mathématicien aux prises avec le siècle. Free translation: «Life is strange. In fact, in geometry, we do not think in the same way of a complex affine line (for example in the theorem of Ceva in a triangle) and of the field of complex numbers x+iy. When I think about this, imaginary points in geomtry are gray, the real points are black, and the intersection of two conjugate imaginary lines is a black real point. The beautiful umbilical conic is silver, the lines and isotropic cones are mostly pink.» 


"God exists since mathematics is consistent, and the Devil exists since we cannot prove it." André Weil 


It's hard to beat John Stembridge's page of quotes. My single favorite one on this page: "If I have not seen as far as others, it is because there were giants standing on my shoulders."  Hal Abelson. 


"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is." Paul Erdős 


“The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.” Godfrey Harold Hardy 


C. G. J. Jacobi, writing to von Humboldt, in 1846. Without pretty ßs: Only Dirichlet, Not I, not Cauchy, not Gauss, knows what a perfectly rigourous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fiftyfifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters. 


D. Hilbert, talking about an exstudent. I'd love to remember where I got this from! 


"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."  Henry Poincaré 


"You don't have to believe in God, but you should believe in The Book."  Paul Erdős. describing the Book held by the God that contains the most beautiful proofs to all the theorems 


"My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful." Herman Weyl 


"A mathematician who is not also something of a poet will never be a perfect mathematician" Karl Weierstraß 


"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions", F. Klein (from Reed & Simon: Methods of modern mathematical physics) 


GianCarlo Rota, in an interview with David Sharp. 


Someone once told me that Grothendieck said "a sheaf of groups is a group of sheaves," although I have been unable to find a real reference. Can anyone substantiate this? 


" Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?" " Serge Lang 


"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." Pierre de Fermat 


"A mathematical truth is neither simple nor complicated in itself, it is."  Émile Lemoine 


In mathematics you don't understand things. You just get used to them. John von Neumann, reply to a physicist at Los Alamos who had said "I don't understand the method of characteristics."  footnote on page 226 of Gary Zukav, The Dancing Wu Li Masters: An Overview of the New Physics, Rider, London, 1990. (taken from Warren Dicks' Home Page) 

